College Algebra Exam Review 65 - concept out of the common...

1.11. RINGS AND FIELDS 75 example, z 7! 2z C 1 3z ± 1=2 maps 1=6 to 1 and 1 to 2=3 . Show that the set of fractional linear transformations is closed under composition. Find the condition for a fractional linear transformation to be invertible, and show that the set of invertible fractional linear transformations forms a group. 1.11. Rings and Fields You are familiar with several algebraic systems having two operations, ad-dition and multiplication, satisfying several of the usual laws of arithmetic: 1. The set Z of integers 2. The set KŒxŁ of polynomials over a ﬁeld K 3. The set Z n of integers modulo n 4. The set Mat n . R / of n-by-n matrices with real entries, with entry-by-entry addition of matrices, and matrix multiplication In each of these examples, the set is group under the operation of ad-dition, multiplication is associative, and there are distributive laws relating multiplication and addition: x.a C b/ D xa C xb , and .a C b/x D ax C bx . In the ﬁrst three examples, multiplication is commutative, but in the fourth example, it is not. Again, it is fruitful to make a

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Unformatted text preview: concept out of the common characteris-tics of these examples, so we make the following denition: Denition 1.11.1. A ring is a (nonempty) set R with two operations: ad-dition, denoted here by C , and multiplication, denoted by juxtaposition, satisfying the following requirements: (a) Under addition, R is an abelian group. (b) Multiplication is associative. (c) Multiplication distributes over addition: a.b C c/ D ab C ac , and .b C c/a D ba C ca for all a;b;c 2 R . Multiplication need not be commutative in a ring, in general. If multi-plication is commutative, the ring is called a commutative ring . Some additional examples of rings are as follows: Example 1.11.2. (a) The familiar number systems R , Q , C are rings. The set of nat-ural numbers N is not a ring, because it is not a group under addition. (b) The set KX;Y of polynomials in two variables over a eld K is a commutative ring....
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